Abstract
LetV andW be two Banach spaces, withV reflexive,
a bounded convex set ofV, A a linear mapping fromV intoW, and letF be a convex functional onW. We minimizeJ(v)=F(Av) on
using hypotheses about particular sequences in
IfV is uniformly convex, we prove existence and uniqueness of a solution of minimal norm minimizingJ. In the Hilbert space case, withF defined byF(w)=∥w−f∥ 2,f given inW, we get existence and uniqueness of the projection off on A(
), which generalizes the case where A(
) is a closed set ofW (taking
closed andA continuous). Finally, we give examples, and we study an unbounded operator case.
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Communicated by A. V. Balakrishnan
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Gioan, A. Regularized minimization under weaker hypotheses. Appl Math Optim 8, 59–67 (1982). https://doi.org/10.1007/BF01447751
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DOI: https://doi.org/10.1007/BF01447751